Diameter and maximum degree in Eulerian digraphs

Dankelmann, Peter; Dorfling, Michael

doi:10.1016/j.disc.2015.11.021

Cited by 7 publications

(5 citation statements)

References 9 publications

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“…by (12). Now +3a+b−1 i= +3a n i is at least 0 if b = 0, at least 1 if b = 1, and at least 3 if b = 2 since (n +3a , n +3a+1 ) is not a 1-pair.…”

Section: Theoremmentioning

confidence: 92%

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Bounds on the fault‐diameter of graphs

Dankelmann

2017

Networks

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Let G be a ( k + 1 ) ‐connected or ( k + 1 ) ‐edge‐connected graph, where k ∈ ℕ . The k‐fault‐diameter and k‐edge‐fault‐diameter of G is the largest diameter of the subgraphs obtained from G by removing up to k vertices and edges, respectively. In this paper we give upper bounds on the k‐fault‐diameter and k‐edge‐fault‐diameter of graphs in terms of order. We show that the k‐fault‐diameter of a ( k + 1 ) ‐connected graph G of order n is bounded from above by n − k + 1 , and by approximately 4 k + 2 n if G is also triangle‐free. If G does not contain 4‐cycles then this bound can be improved further to approximately 5 n ( k − 1 ) 2 . We further show that the k‐edge‐fault‐diameter of a ( k + 1 ) ‐edge‐connected graph of order n is bounded by n – 1 if k = 1, by ⌊ 2 n − 1 3 ⌋ if k = 2, and by approximately 3 k + 2 n if k ≥ 3 , and give improved bounds for triangle‐free graphs. Some of the latter bounds strengthen, in some sense, bounds by Erdös, Pach, Pollack, and Tuza (J Combin Theory B 47 (1989) 73–79) on the diameter. All bounds presented are sharp or at least close to being optimal. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(2), 132–140 2017

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“…by (12). Now +3a+b−1 i= +3a n i is at least 0 if b = 0, at least 1 if b = 1, and at least 3 if b = 2 since (n +3a , n +3a+1 ) is not a 1-pair.…”

Section: Theoremmentioning

confidence: 92%

“…Watkins showed that the diameter of a κ ‐connected graph of order n is bounded by

⌊ \frac{n - 2}{κ} ⌋ + 1

, and this bound is sharp. For more recent bounds on the diameter of graphs and digraphs see, for example, .…”

Section: Introductionmentioning

confidence: 99%

Bounds on the fault‐diameter of graphs

Dankelmann

2017

Networks

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“…Studies have also been carried out regarding the oriented diameter of specific subclasses of graphs like AT-free graphs, interval graphs, chordal graphs and planar graphs [4,5,6]. Bounds on oriented diameter in terms of other graph parameters like minimum degree and maximum degree are also available in literature [7,8,9,10].…”

Section: Introductionmentioning

confidence: 99%

An Improvement to Chvátal and Thomassen's Upper Bound for Oriented Diameter

Babu¹,

Benson²,

Rajendraprasad³

et al. 2020

Preprint

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An orientation of an undirected graph G is an assignment of exactly one direction to each edge of G. The oriented diameter of a graph G is the smallest diameter among all the orientations of G. The maximum oriented diameter of a family of graphs F is the maximum oriented diameter among all the graphs in F . Chvátal and Thomassen [JCTB, 1978] gave a lower bound of 1 2 d 2 + d and an upper bound of 2d 2 + 2d for the maximum oriented diameter of the family of 2-edge connected graphs of diameter d. We improve this upper bound to 1.373d 2 + 6.971d − 1, which outperforms the former upper bound for all values of d greater than or equal to 8. For the family of 2-edge connected graphs of diameter 3, Kwok, Liu and West [JCTB, 2010] obtained improved lower and upper bounds of 9 and 11 respectively. For the family of 2-edge connected graphs of diameter 4, the bounds provided by Chvátal and Thomassen are 12 and 40 and no better bounds were known. By extending the method we used for diameter d graphs, along with an asymmetric extension of a technique used by Chvátal and Thomassen, we have improved this upper bound to 21.

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“…for n = 5) has a vertex whose in‐degree and out‐degree is

0false \frac{n - 1}{2}

. However, it was shown in that, in Eulerian digraphs, a slightly weaker version of holds.…”

Section: Introduction and Terminologymentioning

confidence: 99%

“…1 for n = 5) has a vertex whose in-degree and out-degree is −1 2 . However, it was shown in [6] that, in Eulerian digraphs, a slightly weaker version of 1 holds. In this article, we consider the natural question whether a high-degree vertex in a bridgeless graph also guarantees that the oriented diameter is not too large and answer it in the affirmative.…”

mentioning

confidence: 99%

Oriented diameter of graphs with given maximum degree

Dankelmann

Guo

Surmacs

2017

Journal of Graph Theory

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In this article, we show that every bridgeless graph G of order n and maximum degree Δ has an orientation of diameter at most n−Δ+3. We then use this result and the definition NGfalse(Hfalse)=⋃v∈V(H)NGfalse(vfalse)∖Vfalse(Hfalse), for every subgraph H of G, to give better bounds in the case that G contains certain clusters of high‐degree vertices, namely: For every edge e, G has an orientation of diameter at most n−false|NG(e)false|+4, if e is on a triangle and at most n−false|NG(e)false|+5, otherwise. Furthermore, for every bridgeless subgraph H of G, there is such an orientation of diameter at most n−false|NG(H)false|+3. Finally, if G is bipartite, then we show the existence of an orientation of diameter at most 2(|A|− deg Gfalse(sfalse))+7, for every partite set A of G and s∈V(G)∖A. This particularly implies that balanced bipartite graphs have an orientation of diameter at most n−2Δ+7. For each bound, we give a polynomial‐time algorithm to construct a corresponding orientation and an infinite family of graphs for which the bound is sharp.

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Diameter and maximum degree in Eulerian digraphs

Cited by 7 publications

References 9 publications

Bounds on the fault‐diameter of graphs

Bounds on the fault‐diameter of graphs

An Improvement to Chvátal and Thomassen's Upper Bound for Oriented Diameter

Oriented diameter of graphs with given maximum degree

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